Print all possible combinations of r elements in a given array of size n

Given an array of size n, generate and print all possible combinations of r elements in array. For example, if input array is {1, 2, 3, 4} and r is 2, then output should be {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4} and {3, 4}.
Following are two methods to do this. 
Method 1 (Fix Elements and Recur)
We create a temporary array ‘data[]’ which stores all outputs one by one. The idea is to start from first index (index = 0) in data[], one by one fix elements at this index and recur for remaining indexes. Let the input array be {1, 2, 3, 4, 5} and r be 3. We first fix 1 at index 0 in data[], then recur for remaining indexes, then we fix 2 at index 0 and recur. Finally, we fix 3 and recur for remaining indexes. When number of elements in data[] becomes equal to r (size of a combination), we print data[].
Following diagram shows recursion tree for same input. 

Following is the implementation of the above approach.  

C++

// C++ program to print all combination
// of size r in an array of size n
#include<bits/stdc++.h>
using namespace std;

void combinationUtil(int arr[], int data[],
                    int start, int end,
                    int index, int r);

// The main function that prints
// all combinations of size r
// in arr[] of size n. This function
// mainly uses combinationUtil()
void printCombination(int arr[], int n, int r)
{
    // A temporary array to store
    // all combination one by one
    int data[r];

    // Print all combination using
    // temporary array 'data[]'
    combinationUtil(arr, data, 0, n-1, 0, r);
}

/* arr[] ---> Input Array
data[] ---> Temporary array to
store current combination
start & end ---> Starting and
Ending indexes in arr[]
index ---> Current index in data[]
r ---> Size of a combination to be printed */
void combinationUtil(int arr[], int data[],
                    int start, int end,
                    int index, int r)
{
    // Current combination is ready
    // to be printed, print it
    if (index == r)
    {
        for (int j = 0; j < r; j++)
            cout << data[j] << " ";
        cout << endl;
        return;
    }

    // replace index with all possible
    // elements. The condition "end-i+1 >= r-index"
    // makes sure that including one element
    // at index will make a combination with
    // remaining elements at remaining positions
    for (int i = start; i <= end &&
        end - i + 1 >= r - index; i++)
    {
        data[index] = arr[i];
        combinationUtil(arr, data, i+1,
                        end, index+1, r);
    }
}

// Driver code
int main()
{
    int arr[] = {1, 2, 3, 4, 5};
    int r = 3;
    int n = sizeof(arr)/sizeof(arr[0]);
    printCombination(arr, n, r);
}

// This code is contributed by rathbhupendra

C

// Program to print all combination of size r in an array of size n
#include <stdio.h>
void combinationUtil(int arr[], int data[], int start, int end,
                     int index, int r);

// The main function that prints all combinations of size r
// in arr[] of size n. This function mainly uses combinationUtil()
void printCombination(int arr[], int n, int r)
{
    // A temporary array to store all combination one by one
    int data[r];

    // Print all combination using temporary array 'data[]'
    combinationUtil(arr, data, 0, n-1, 0, r);
}

/* arr[]  ---> Input Array
   data[] ---> Temporary array to store current combination
   start & end ---> Starting and Ending indexes in arr[]
   index  ---> Current index in data[]
   r ---> Size of a combination to be printed */
void combinationUtil(int arr[], int data[], int start, int end,
                     int index, int r)
{
    // Current combination is ready to be printed, print it
    if (index == r)
    {
        for (int j=0; j<r; j++)
            printf("%d ", data[j]);
        printf("\n");
        return;
    }

    // replace index with all possible elements. The condition
    // "end-i+1 >= r-index" makes sure that including one element
    // at index will make a combination with remaining elements
    // at remaining positions
    for (int i=start; i<=end && end-i+1 >= r-index; i++)
    {
        data[index] = arr[i];
        combinationUtil(arr, data, i+1, end, index+1, r);
    }
}

// Driver program to test above functions
int main()
{
    int arr[] = {1, 2, 3, 4, 5};
    int r = 3;
    int n = sizeof(arr)/sizeof(arr[0]);
    printCombination(arr, n, r);
}

Java

// Java program to print all combination of size r in an array of size n
import java.io.*;

class Combination {

    /* arr[]  ---> Input Array
    data[] ---> Temporary array to store current combination
    start & end ---> Starting and Ending indexes in arr[]
    index  ---> Current index in data[]
    r ---> Size of a combination to be printed */
    static void combinationUtil(int arr[], int data[], int start,
                                int end, int index, int r)
    {
        // Current combination is ready to be printed, print it
        if (index == r)
        {
            for (int j=0; j<r; j++)
                System.out.print(data[j]+" ");
            System.out.println("");
            return;
        }

        // replace index with all possible elements. The condition
        // "end-i+1 >= r-index" makes sure that including one element
        // at index will make a combination with remaining elements
        // at remaining positions
        for (int i=start; i<=end && end-i+1 >= r-index; i++)
        {
            data[index] = arr[i];
            combinationUtil(arr, data, i+1, end, index+1, r);
        }
    }

    // The main function that prints all combinations of size r
    // in arr[] of size n. This function mainly uses combinationUtil()
    static void printCombination(int arr[], int n, int r)
    {
        // A temporary array to store all combination one by one
        int data[]=new int[r];

        // Print all combination using temporary array 'data[]'
        combinationUtil(arr, data, 0, n-1, 0, r);
    }

    /*Driver function to check for above function*/
    public static void main (String[] args) {
        int arr[] = {1, 2, 3, 4, 5};
        int r = 3;
        int n = arr.length;
        printCombination(arr, n, r);
    }
}

/* This code is contributed by Devesh Agrawal */

Python3

# Program to print all combination
# of size r in an array of size n

# The main function that prints
# all combinations of size r in
# arr[] of size n. This function
# mainly uses combinationUtil()
def printCombination(arr, n, r):

    # A temporary array to
    # store all combination
    # one by one
    data = [0]*r;

    # Print all combination
    # using temporary array 'data[]'
    combinationUtil(arr, data, 0,
                    n - 1, 0, r);

# arr[] ---> Input Array
# data[] ---> Temporary array to
#         store current combination
# start & end ---> Starting and Ending
#             indexes in arr[]
# index ---> Current index in data[]
# r ---> Size of a combination
# to be printed
def combinationUtil(arr, data, start,
                    end, index, r):

    # Current combination is ready
    # to be printed, print it
    if (index == r):
        for j in range(r):
            print(data[j], end = " ");
        print();
        return;

    # replace index with all
    # possible elements. The
    # condition "end-i+1 >=
    # r-index" makes sure that
    # including one element at
    # index will make a combination
    # with remaining elements at
    # remaining positions
    i = start;
    while(i <= end and end - i + 1 >= r - index):
        data[index] = arr[i];
        combinationUtil(arr, data, i + 1,
                        end, index + 1, r);
        i += 1;

# Driver Code
arr = [1, 2, 3, 4, 5];
r = 3;
n = len(arr);
printCombination(arr, n, r);

# This code is contributed by mits

C#

// C# program to print all
// combination of size r
// in an array of size n
using System;

class GFG
{
    /* arr[] ---> Input Array
    data[] ---> Temporary array to
                store current combination
    start & end ---> Starting and Ending
                     indexes in arr[]
    index ---> Current index in data[]
    r ---> Size of a combination
            to be printed */
    static void combinationUtil(int []arr, int []data,
                                int start, int end,
                                int index, int r)
    {
        // Current combination is
        // ready to be printed,
        // print it
        if (index == r)
        {
            for (int j = 0; j < r; j++)
                Console.Write(data[j] + " ");
            Console.WriteLine("");
            return;
        }

        // replace index with all
        // possible elements. The
        // condition "end-i+1 >=
        // r-index" makes sure that
        // including one element
        // at index will make a
        // combination with remaining
        // elements at remaining positions
        for (int i = start; i <= end &&
                  end - i + 1 >= r - index; i++)
        {
            data[index] = arr[i];
            combinationUtil(arr, data, i + 1,
                            end, index + 1, r);
        }
    }

    // The main function that prints
    // all combinations of size r
    // in arr[] of size n. This
    // function mainly uses combinationUtil()
    static void printCombination(int []arr,
                                 int n, int r)
    {
        // A temporary array to store
        // all combination one by one
        int []data = new int[r];

        // Print all combination
        // using temporary array 'data[]'
        combinationUtil(arr, data, 0,
                        n - 1, 0, r);
    }

    // Driver Code
    static public void Main ()
    {
        int []arr = {1, 2, 3, 4, 5};
        int r = 3;
        int n = arr.Length;
        printCombination(arr, n, r);
    }
}

// This code is contributed by m_kit

PHP

<?php
// Program to print all
// combination of size r
// in an array of size n

// The main function that
// prints all combinations
// of size r in arr[] of
// size n. This function
// mainly uses combinationUtil()
function printCombination($arr,
                          $n, $r)
{
    // A temporary array to
    // store all combination
    // one by one
    $data = array();

    // Print all combination
    // using temporary array 'data[]'
    combinationUtil($arr, $data, 0,
                    $n - 1, 0, $r);
}

/* arr[] ---> Input Array
data[] ---> Temporary array to
            store current combination
start & end ---> Starting and Ending
                 indexes in arr[]
index ---> Current index in data[]
r ---> Size of a combination
       to be printed */
function combinationUtil($arr, $data, $start,
                         $end, $index, $r)

{
    // Current combination is ready
    // to be printed, print it
    if ($index == $r)
    {
        for ($j = 0; $j < $r; $j++)
            echo $data[$j];
        echo "\n";
        return;
    }

    // replace index with all
    // possible elements. The
    // condition "end-i+1 >=
    // r-index" makes sure that
    // including one element at
    // index will make a combination
    // with remaining elements at
    // remaining positions
    for ($i = $start;
         $i <= $end &&
         $end - $i + 1 >= $r - $index; $i++)
    {
        $data[$index] = $arr[$i];
        combinationUtil($arr, $data, $i + 1,
                        $end, $index + 1, $r);
    }
}

// Driver Code
$arr = array(1, 2, 3, 4, 5);
$r = 3;
$n = sizeof($arr);
printCombination($arr, $n, $r);

// This code is contributed by ajit
?>

Javascript

<script>

// Javascript program to print all
// combination of size r in an array of size n

    /* arr[]  ---> Input Array
    data[] ---> Temporary array to store current combination
    start & end ---> Starting and Ending indexes in arr[]
    index  ---> Current index in data[]
    r ---> Size of a combination to be printed */
    function combinationUtil(arr,data,start,end,index,r)
    {
        // Current combination is ready to be printed, print it
        if (index == r)
        {
            for (let j=0; j<r; j++)
            {
                document.write(data[j]+" ");
            }
            document.write("<br>")
        }

        // replace index with all possible elements. The condition
        // "end-i+1 >= r-index" makes sure that including one element
        // at index will make a combination with remaining elements
        // at remaining positions
        for (let i=start; i<=end && end-i+1 >= r-index; i++)
        {
            data[index] = arr[i];
            combinationUtil(arr, data, i+1, end, index+1, r);
        }
    }

    // The main function that prints all combinations of size r
    // in arr[] of size n. This function mainly uses combinationUtil()
    function printCombination(arr,n,r)
    {
        // A temporary array to store all combination one by one
        let data = new Array(r);

        // Print all combination using temporary array 'data[]'
        combinationUtil(arr, data, 0, n-1, 0, r);
    }

    /*Driver function to check for above function*/
    let arr=[1, 2, 3, 4, 5];
    let r = 3;
    let n = arr.length;
    printCombination(arr, n, r);


    // This code is contributed by rag2127

</script>

Output

1 2 3 
1 2 4 
1 2 5 
1 3 4 
1 3 5 
1 4 5 
2 3 4 
2 3 5 
2 4 5 
3 4 5 

Time Complexity: O(n^2)
Auxiliary Space: O(r). We use a temporary array data[] of size r to store current combination.

How to handle duplicates?
Note that the above method doesn’t handle duplicates. For example, if input array is {1, 2, 1} and r is 2, then the program prints {1, 2} and {2, 1} as two different combinations. We can avoid duplicates by adding following two additional things to above code. 
1) Add code to sort the array before calling combinationUtil() in printCombination() 
2) Add following lines at the end of for loop in combinationUtil() 

        // Since the elements are sorted, all occurrences of an element
        // must be together
        while (arr[i] == arr[i+1])
             i++; 

See this for an implementation that handles duplicates.
Method 2 (Include and Exclude every element)
Like the above method, We create a temporary array data[]. The idea here is similar to Subset Sum Problem. We one by one consider every element of input array, and recur for two cases:
1) The element is included in current combination (We put the element in data[] and increment next available index in data[]) 
2) The element is excluded in current combination (We do not put the element and do not change index)
When number of elements in data[] become equal to r (size of a combination), we print it.
This method is mainly based on Pascal’s Identity, i.e. n_cr = n-1_cr + n-1_cr-1
Following is implementation of method 2. 

C++

// C++ Program to print all combination of
// size r in an array of size n
#include <bits/stdc++.h>
using namespace std;
void combinationUtil(int arr[], int n, int r,
                    int index, int data[], int i);

// The main function that prints all
// combinations of size r in arr[]
// of size n. This function mainly
// uses combinationUtil()
void printCombination(int arr[], int n, int r)
{
    // A temporary array to store
    // all combination one by one
    int data[r];

    // Print all combination using
    // temporary array 'data[]'
    combinationUtil(arr, n, r, 0, data, 0);
}

/* arr[] ---> Input Array
n ---> Size of input array
r ---> Size of a combination to be printed
index ---> Current index in data[]
data[] ---> Temporary array to store current combination
i ---> index of current element in arr[] */
void combinationUtil(int arr[], int n, int r,
                    int index, int data[], int i)
{
    // Current combination is ready, print it
    if (index == r)
    {
        for (int j = 0; j < r; j++)
            cout << data[j] << " ";
        cout << endl;
        return;
    }

    // When no more elements are there to put in data[]
    if (i >= n)
        return;

    // current is included, put next at next location
    data[index] = arr[i];
    combinationUtil(arr, n, r, index + 1, data, i + 1);

    // current is excluded, replace it with next (Note that
    // i+1 is passed, but index is not changed)
    combinationUtil(arr, n, r, index, data, i+1);
}

// Driver code
int main()
{
    int arr[] = {1, 2, 3, 4, 5};
    int r = 3;
    int n = sizeof(arr)/sizeof(arr[0]);
    printCombination(arr, n, r);
    return 0;
}

// This is code is contributed by rathbhupendra

C

// Program to print all combination of size r in an array of size n
#include<stdio.h>
void combinationUtil(int arr[],int n,int r,int index,int data[],int i);

// The main function that prints all combinations of size r
// in arr[] of size n. This function mainly uses combinationUtil()
void printCombination(int arr[], int n, int r)
{
    // A temporary array to store all combination one by one
    int data[r];

    // Print all combination using temporary array 'data[]'
    combinationUtil(arr, n, r, 0, data, 0);
}

/* arr[]  ---> Input Array
   n      ---> Size of input array
   r      ---> Size of a combination to be printed
   index  ---> Current index in data[]
   data[] ---> Temporary array to store current combination
   i      ---> index of current element in arr[]     */
void combinationUtil(int arr[], int n, int r, int index, int data[], int i)
{
    // Current combination is ready, print it
    if (index == r)
    {
        for (int j=0; j<r; j++)
            printf("%d ",data[j]);
        printf("\n");
        return;
    }

    // When no more elements are there to put in data[]
    if (i >= n)
        return;

    // current is included, put next at next location
    data[index] = arr[i];
    combinationUtil(arr, n, r, index+1, data, i+1);

    // current is excluded, replace it with next (Note that
    // i+1 is passed, but index is not changed)
    combinationUtil(arr, n, r, index, data, i+1);
}

// Driver program to test above functions
int main()
{
    int arr[] = {1, 2, 3, 4, 5};
    int r = 3;
    int n = sizeof(arr)/sizeof(arr[0]);
    printCombination(arr, n, r);
    return 0;
}

Java

// Java program to print all combination of size r in an array of size n
import java.io.*;

class Combination {

    /* arr[]  ---> Input Array
    data[] ---> Temporary array to store current combination
    start & end ---> Starting and Ending indexes in arr[]
    index  ---> Current index in data[]
    r ---> Size of a combination to be printed */
    static void combinationUtil(int arr[], int n, int r, int index,
                                int data[], int i)
    {
        // Current combination is ready to be printed, print it
        if (index == r)
        {
            for (int j=0; j<r; j++)
                System.out.print(data[j]+" ");
            System.out.println("");
        return;
        }

        // When no more elements are there to put in data[]
        if (i >= n)
        return;

        // current is included, put next at next location
        data[index] = arr[i];
        combinationUtil(arr, n, r, index+1, data, i+1);

        // current is excluded, replace it with next (Note that
        // i+1 is passed, but index is not changed)
        combinationUtil(arr, n, r, index, data, i+1);
    }

    // The main function that prints all combinations of size r
    // in arr[] of size n. This function mainly uses combinationUtil()
    static void printCombination(int arr[], int n, int r)
    {
        // A temporary array to store all combination one by one
        int data[]=new int[r];

        // Print all combination using temporary array 'data[]'
        combinationUtil(arr, n, r, 0, data, 0);
    }

    /*Driver function to check for above function*/
    public static void main (String[] args) {
        int arr[] = {1, 2, 3, 4, 5};
        int r = 3;
        int n = arr.length;
        printCombination(arr, n, r);
    }
}
/* This code is contributed by Devesh Agrawal */

Python 3

# Program to print all combination
# of size r in an array of size n

# The main function that prints all
# combinations of size r in arr[] of
# size n. This function mainly uses
# combinationUtil()
def printCombination(arr, n, r):

    # A temporary array to store
    # all combination one by one
    data = [0] * r

    # Print all combination using
    # temporary array 'data[]'
    combinationUtil(arr, n, r, 0, data, 0)

''' arr[] ---> Input Array
n     ---> Size of input array
r     ---> Size of a combination to be printed
index ---> Current index in data[]
data[] ---> Temporary array to store
            current combination
i     ---> index of current element in arr[]     '''
def combinationUtil(arr, n, r, index, data, i):

    # Current combination is ready,
    # print it
    if (index == r):
        for j in range(r):
            print(data[j], end = " ")
        print()
        return

    # When no more elements are
    # there to put in data[]
    if (i >= n):
        return

    # current is included, put
    # next at next location
    data[index] = arr[i]
    combinationUtil(arr, n, r, index + 1,
                    data, i + 1)

    # current is excluded, replace it
    # with next (Note that i+1 is passed,
    # but index is not changed)
    combinationUtil(arr, n, r, index,
                    data, i + 1)

# Driver Code
if __name__ == "__main__":
    arr = [1, 2, 3, 4, 5]
    r = 3
    n = len(arr)
    printCombination(arr, n, r)

# This code is contributed
# by ChitraNayal

C#

// C# program to print all
// combination of size r
// in an array of size n
using System;

class GFG
{

    /* arr[] ---> Input Array
    data[] ---> Temporary array to
                store current combination
    start & end ---> Starting and Ending
                     indexes in arr[]
    index ---> Current index in data[]
    r ---> Size of a combination
           to be printed */
    static void combinationUtil(int []arr, int n,
                                int r, int index,
                                int []data, int i)
    {
        // Current combination is ready
        // to be printed, print it
        if (index == r)
        {
            for (int j = 0; j < r; j++)
                Console.Write(data[j] + " ");
                Console.WriteLine("");
            return;
        }

        // When no more elements are
        // there to put in data[]
        if (i >= n)
        return;

        // current is included, put
        // next at next location
        data[index] = arr[i];
        combinationUtil(arr, n, r,
                        index + 1, data, i + 1);

        // current is excluded, replace
        // it with next (Note that
        // i+1 is passed, but index
        // is not changed)
        combinationUtil(arr, n, r, index,
                        data, i + 1);
    }

    // The main function that prints
    // all combinations of size r
    // in arr[] of size n. This
    // function mainly uses combinationUtil()
    static void printCombination(int []arr,
                                 int n, int r)
    {
        // A temporary array to store
        // all combination one by one
        int []data = new int[r];

        // Print all combination
        // using temporary array 'data[]'
        combinationUtil(arr, n, r, 0, data, 0);
    }

    // Driver Code
    static public void Main ()
    {
        int []arr = {1, 2, 3, 4, 5};
        int r = 3;
        int n = arr.Length;
        printCombination(arr, n, r);
    }
}

// This code is contributed by ajit

PHP

<?php
// Program to print all
// combination of size r
// in an array of size n

// The main function that prints
// all combinations of size r in
// arr[] of size n. This function
// mainly uses combinationUtil()
function printCombination($arr, $n, $r)
{
    // A temporary array to store
    // all combination one by one
    $data = Array();

    // Print all combination using
    // temporary array 'data[]'
    combinationUtil($arr, $n, $r,
                    0, $data, 0);
}

/* arr[] ---> Input Array
n ---> Size of input array
r ---> Size of a combination
       to be printed
index ---> Current index in data[]
data[] ---> Temporary array to store
            current combination
i ---> index of current element in arr[] */
function combinationUtil($arr, $n, $r,
                         $index, $data, $i)
{
    // Current combination
    // is ready, print it
    if ($index == $r)
    {
        for ($j = 0; $j < $r; $j++)
            echo $data[$j], " ";
        echo "\n";
        return;
    }

    // When no more elements are
    // there to put in data[]
    if ($i >= $n)
        return;

    // current is included, put
    // next at next location
    $data[$index] = $arr[$i];
    combinationUtil($arr, $n, $r,
                    $index + 1,
                    $data, $i + 1);

    // current is excluded, replace
    // it with next (Note that i+1
    // is passed, but index is not changed)
    combinationUtil($arr, $n, $r,
                    $index, $data, $i + 1);
}

// Driver Code
$arr = array(1, 2, 3, 4, 5);
$r = 3;
$n = sizeof($arr);
printCombination($arr, $n, $r);

// This code is contributed by ajit
?>

Javascript

<script>

// Javascript program to print all
// combination of size r in an array of size n

    /* arr[]  ---> Input Array
    data[] ---> Temporary array to
    store current combination
    start & end ---> Starting and
    Ending indexes in arr[]
    index  ---> Current index in data[]
    r ---> Size of a combination to be printed */
    function combinationUtil(arr,n,r,index,data,i)
    {
        // Current combination is ready
        // to be printed, print it
        if (index == r)
        {
            for (let j=0; j<r; j++)
            {
                document.write(data[j]+" ");
            }
            document.write("<br>");

            return;
        }

        // When no more elements are there
        // to put in data[]
        if (i >= n)
        {
            return;
        }

        // current is included, put
        // next at next location
        data[index] = arr[i];
        combinationUtil(arr, n, r, index+1, data, i+1);

        // current is excluded, replace
        // it with next (Note that
        // i+1 is passed, but index is not changed)
        combinationUtil(arr, n, r, index, data, i+1);

    }

    // The main function that prints
    // all combinations of size r
    // in arr[] of size n. This function
    // mainly uses combinationUtil()
    function printCombination(arr,n,r)
    {
        // A temporary array to store
        // all combination one by one
        let data=new Array(r);

        // Print all combination using
        // temporary array 'data[]'
        combinationUtil(arr, n, r, 0, data, 0);
    }

    /*Driver function to check for above function*/
    let arr=[1, 2, 3, 4, 5];
    let r = 3;
    let n = arr.length;
    printCombination(arr, n, r);

    // This code is contributed by avanitrachhadiya2155

</script>

Output

1 2 3 
1 2 4 
1 2 5 
1 3 4 
1 3 5 
1 4 5 
2 3 4 
2 3 5 
2 4 5 
3 4 5 

Time Complexity: O(n^r)
Auxiliary Space : O(r)

How to handle duplicates in method 2?
Like method 1, we can following two things to handle duplicates. 
1) Add code to sort the array before calling combinationUtil() in printCombination() 
2) Add following lines between two recursive calls of combinationUtil() in combinationUtil() 

        // Since the elements are sorted, all occurrences of an element
        // must be together
        while (arr[i] == arr[i+1])
             i++; 

See this for an implementation that handles duplicates.
Below is another DFS based approach to solve this problem. 
Make all combinations of size k
This article is contributed by Bateesh. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above